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Functional equations, joinings of skew products, and ergodic theorems for diagonal measures. (Équations fonctionnelles, couplages de produits gauches et théorèmes ergodiques pour mesures diagonales.) (French) Zbl 0798.28008
This interesting paper looks at certain ergodic properties of compact Abelian extensions of discrete spectrum systems and applies the results to obtain multiple recurrence type theorems.
Let $$T:X \to X$$; $$Tx=x + \alpha$$, be an ergodic rotation on a compact Abelian metrizable group. If $$(G,\cdot)$$ is another such group, then a measurable function $$\varphi:X \to G$$ is called a cocycle. The corresponding group extension is $S_ \varphi = (X \times G,{\mathcal B} (X) \otimes {\mathcal B}(G),\;m_ X \otimes m_ G,T_ \varphi),$ defined by $$T_ \varphi(x,g)= (x + \alpha,g \cdot \varphi(x))$$, where $${\mathcal B} (\cdot)$$ denotes the $$\sigma$$-algebra of Borel sets, and $$m_{(\cdot)}$$ is Haar measure. The cocycle $$\varphi$$ is called a coboundary if $$\exists$$ a measurable function $$\psi:X \to G$$ satisfying $$\varphi = \psi \circ T/ \psi$$, and it is a quasi-coboundary if $$\exists g \in G$$ such that $$\varphi \cdot g$$ is a coboundary. Finally, $$\varphi$$ is said to be irreducible if $$\forall \sigma \in \widehat G$$, the character group of $$G$$, $$(\sigma \neq 1)$$, the set $\bigl \{t \in X: \sigma \circ \varphi (x+t)/ \sigma \circ \varphi(x) \text{ is a quasi-coboundary (from } X \text{ to }S^ 1) \bigr\}$ is null $$(S^ 1=$$ unit circle in the complex plane).
It is shown that if $$\varphi:X \to S^ 1$$ is a cocycle for which there exists a non-null set of values $$t\in X$$ for which the cocycle $$\varphi(x+t)/ \varphi(x)$$ is a quasi-coboundary, then $$S_ \varphi$$ is a factor of a translation on a compact quotient of a nilpotent group. In particular, if $$X$$ is a 1-dimensional torus, this condition on $$\varphi$$ ensures that $$S_ \varphi$$ has quasi-discrete spectrum.
The following are the main theorems of the paper:
Theorem 1. Let $$(\varphi_ n)_{n \in \mathbb{Z}}$$ be a sequence of cocycles from $$X \to G$$. For each $$t \in X$$ we associate a cocycle $$\varphi_ t^ \mathbb{Z}:X \to G^ \mathbb{Z}$$; $$\varphi_ t^ \mathbb{Z}(x) = (\varphi_ n(x+nt))_{n \in \mathbb{Z}}$$. If all the $$\varphi_ n$$’s are irreducible then, for all $$t$$, the cocycle $$\varphi_ t^ \mathbb{Z}$$ is weakly mixing.
Theorem 2. Let $$p \in \mathbb{Z}^ +$$, and $$(\varphi_ j)^ p_{j=-p}$$ be cocycles from $$X \to G$$, $$T_ j = T_{\varphi_ j}$$. Then for all $$f_{-p}$$, $$f_{-p+1},\dots,f_ p \in L^ \infty (X \times G)$$ a.e. $$(x,g)$$ ${1\over n} \sum^{n-1}_{k=0} \prod^ p_{j=-p} f_ j(T_ j^{jk} (x,g))$ is convergent.

MSC:
 28D05 Measure-preserving transformations
Full Text:
References:
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